(*
  DaoMath Innate Balance Theory
  ±0 先天均衡的形式化定义
*)

Require Import Reals.
Require Import Psatz.
Open Scope R_scope.

(** * 先天均衡的三元结构 *)

(** 先天均衡记录类型 *)
Record InnateBalance : Type := mkBalance {
  positive : R;        (* 正向势能 +1 *)
  negative : R;        (* 负向势能 -1 *)
  neutral : R;         (* 中性态 0 *)
}.

(** 先天均衡的标准实例 *)
Definition standard_balance : InnateBalance :=
  mkBalance 1 (-1) 0.

(** * 先天性质（公理） *)

(** 公理 1: 和为零（总体守恒） *)
Axiom innate_sum_zero : forall (b : InnateBalance),
  positive b + negative b + neutral b = 0.

(** 公理 2: 乘积对偶性 *)
Axiom innate_product_duality : forall (b : InnateBalance),
  positive b * negative b = -1.

(** 公理 3: 存在性 *)
Axiom innate_existence : exists (b : InnateBalance), True.

(** 公理 4: 唯一性（在同构意义下） *)
Axiom innate_uniqueness : forall (b1 b2 : InnateBalance),
  positive b1 + negative b1 = 0 ->
  positive b2 + negative b2 = 0 ->
  positive b1 * negative b1 = positive b2 * negative b2.

(** * 基本定理 *)

(** 定理: 标准均衡满足和为零 *)
Theorem standard_balance_sum_zero :
  positive standard_balance + 
  negative standard_balance + 
  neutral standard_balance = 0.
Proof.
  unfold standard_balance. simpl.
  lra. (* linear real arithmetic *)
Qed.

(** 定理: 标准均衡满足乘积对偶性 *)
Theorem standard_balance_product_duality :
  positive standard_balance * 
  negative standard_balance = -1.
Proof.
  unfold standard_balance. simpl.
  lra.
Qed.

(** 定理: 平衡态的自对偶性 *)
Theorem neutral_self_dual : forall (b : InnateBalance),
  neutral b = - (neutral b) -> neutral b = 0.
Proof.
  intros b H.
  assert (2 * neutral b = 0) by lra.
  lra.
Qed.

(** 定理: 非零平衡态不可能 *)
Theorem nonzero_neutral_impossible : forall (b : InnateBalance),
  neutral b <> 0 ->
  positive b + negative b + neutral b <> 0.
Proof.
  intros b H_nz H_sum.
  apply innate_sum_zero in H_sum.
  contradiction.
Qed.

(** * 对偶变换 *)

(** 对偶操作：交换正负 *)
Definition dual_balance (b : InnateBalance) : InnateBalance :=
  mkBalance (negative b) (positive b) (neutral b).

(** 定理: 对偶的对偶是自身 *)
Theorem dual_involutive : forall (b : InnateBalance),
  dual_balance (dual_balance b) = b.
Proof.
  intros b.
  unfold dual_balance. simpl.
  destruct b. simpl. reflexivity.
Qed.

(** 定理: 对偶保持和为零 *)
Theorem dual_preserves_sum : forall (b : InnateBalance),
  positive b + negative b + neutral b = 0 ->
  positive (dual_balance b) + 
  negative (dual_balance b) + 
  neutral (dual_balance b) = 0.
Proof.
  intros b H.
  unfold dual_balance. simpl.
  lra.
Qed.

(** * 平衡态的线性组合 *)

(** 两个平衡态的组合 *)
Definition combine_balance 
  (b1 b2 : InnateBalance) 
  (alpha beta : R) : InnateBalance :=
  mkBalance 
    (alpha * positive b1 + beta * positive b2)
    (alpha * negative b1 + beta * negative b2)
    (alpha * neutral b1 + beta * neutral b2).

(** 定理: 组合保持守恒（当系数和为1） *)
Theorem combine_preserves_conservation :
  forall (b1 b2 : InnateBalance) (alpha beta : R),
  alpha + beta = 1 ->
  positive b1 + negative b1 + neutral b1 = 0 ->
  positive b2 + negative b2 + neutral b2 = 0 ->
  positive (combine_balance b1 b2 alpha beta) +
  negative (combine_balance b1 b2 alpha beta) +
  neutral (combine_balance b1 b2 alpha beta) = 0.
Proof.
  intros b1 b2 alpha beta H_coeff H1 H2.
  unfold combine_balance. simpl.
  assert (alpha * (positive b1 + negative b1 + neutral b1) = 0) by lra.
  assert (beta * (positive b2 + negative b2 + neutral b2) = 0) by lra.
  lra.
Qed.

(** * 从 ±0 到自然数的构造 *)

(** 后继函数（抽象） *)
Parameter successor : R -> R.

(** 后继公理 *)
Axiom successor_def : forall n, successor n = n + 1.

(** 零元来自中性态 *)
Definition zero_from_balance (b : InnateBalance) : R :=
  neutral b.

(** 单位元来自正向 *)
Definition one_from_balance (b : InnateBalance) : R :=
  abs (positive b).

(** 定理: 标准均衡产生标准数 *)
Theorem standard_numbers_from_balance :
  zero_from_balance standard_balance = 0 /\
  one_from_balance standard_balance = 1.
Proof.
  unfold zero_from_balance, one_from_balance, standard_balance.
  simpl. split.
  - reflexivity.
  - unfold abs. simpl. 
    destruct (Rcase_abs 1); lra.
Qed.

